Method

Set out two bowls. Crack the egg gently on a flat surface or on the rim of a bowl, as close to the middle of the egg as possible. If you crack it on the rim of a bowl it might be easier to get the egg to crack right in the middle, but you are more likely to get pieces of egg shell in the egg whites. These days I crack eggs on a flat surface.

Working over a small bowl, use your thumbs to gently pry the egg halves apart. Let the yolk settle in the lower half of the egg shell while the egg whites run off the sides of the egg into the bowl. Gently transfer the egg yolk back and forth between the egg shell halves, letting as much egg white as you can drip into the bowl below. Be careful so as not to break the egg yolk. Place the egg yolk in a separate bowl.

If you are planning to whip the egg whites for a recipe, you might want to separate the eggs one by one into a smaller bowl, and then transfer the separated egg into larger bowls. This way if you break a yolk it will not break into all the egg whites you've separated. The fat in the egg yolk (or any oil) will interfere with the egg white's ability to whip up properly. For this reason also you should also wash your hands carefully, to remove any natural body oils, before working with egg whites.

If you get a piece of egg shell in the separated eggs, fish it out with a larger piece of shell.

Note that chilled eggs are easier to separate (the yolk doesn't break as easily), but most recipes call for working with eggs at room temperature. So, you either let your eggs get to room temperature before separating them, in which case you'll need to be a bit more careful with the egg yolks, or let the eggs get to room temperature after you've separated them, in which case you should cover them in their bowl with plastic wrap and use them as soon as they get to room temp.